Geometric integrators are presented for a class of nonlinear dispersive equations which includes the Camassa-Holm equation, the BBM equation and the hyperelasticrod wave equation. One group of schemes is designed to preserve a global property of the equations: the conservation of energy; while the other one preserves a more local feature of the equations: the multi-symplecticity.

Spring-based artificial reverberation was one of the earliest attempts at compact replication of room-like reverberation for studio use. The popularity and unique sound of this effect have given it a status and desirability apart from its original use. Standard methods for modeling analog audio effects are not well suited to modeling spring reverberation, due to the complex and dispersive natur...

Problems of linear time-dependent dispersive waves in an unbounded domain are considered. The infinite domain is truncated via an artificial boundary B, and a high-order non-reflecting boundary condition (NRBC) is imposed on B. Then the problem is solved by a finite difference (FD) scheme in the finite domain bounded by B. The sequence of NRBCs proposed by Higdon is used. However, in contrast t...

Electromagnetic wave properties in metamaterials including electromagnetic band gap (EBG) structures have been widely investigated due to their inherent capabilities for developing novel devices in the fields of optics, microwave and antenna engineering. In reality, most of the materials are dispersive or frequency-dependent, on the other hand, it is important to investigate how the performance...

In this article we report on recent work on building numerical approximation schemes for nonlinear Schrödinger equations. We first consider finite-difference space semi-discretizations and show that the standard conservative scheme does not reproduce at the discrete level the dispersion properties of the continuous Schrödinger equation. This is due to high frequency numerical spurious solutions...

Nonstandard finite difference schemes for the Black-Scholes partial differential equation preserving the positivity property are proposed. Computationally simple schemes are derived by using a nonlocal approximation in the reaction term of the Black-Scholes equation. Unlike the standard methods, the solutions of new proposed schemes are positive and free of the spurious oscillations.

It was generally expected that monotone schemes are oscillationfree for hyperbolic conservation laws. However, recently local oscillations were observed and usually understood to be caused by relative phase errors. In order to further explain this, we first investigate the discretization of initial data that trigger the chequerboard mode, the highest frequency mode. Then we proceed to use the d...

We propose a novel finite-difference time-domain (FDTD) scheme for the solution of the Maxwell’s equations in which linear dispersive effects are present. The method uses high-order accurate approximations in space and time for the dispersive Maxwell’s equations written as a second-order vector wave equation with a time-history convolution term. The modified equation approach is combined with t...